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Accepted for publication in Journal of Turbulence 1

Eﬀects of form-induced velocity in rough-wall

turbulent channel ﬂows

By S. C. Mangavelli and J. Yuan†‡

Abstract

Wall roughness induces form-induced (or dispersive) velocity and pressure perturba-

tions inside the roughness sublayer of a wall-bounded turbulent ﬂow. This work discusses

the role played by the form-induced velocity in inﬂuencing turbulence statistics and struc-

ture, using existing direct numerical simulation data of transient half channels in response

to an impulse acceleration (Mangavelli et al.,J. Turbul.,22:434–460, 2021). Focuses are

given to (i) reshaping of turbulent coherent motions by the rate-of-strain of the mean

velocity, and (ii) contributions of diﬀerent velocity sources to turbulent pressure ﬂuctua-

tions. Half-channel ﬂows in both fully-developed and non-equilibrium, transient states are

discussed. Results show that form-induced velocity gradients not only form an important

source of turbulent pressure in an equilibrium ﬂow, but also lead to turbulence produc-

tion and potentially direct structural change of turbulent eddies in a non-equilibrium

ﬂow under acceleration.

1. Introduction

Wall roughness plays an important role in many ﬁelds of study. Much has been done

to identify the eﬀects of roughness on wall-bounded turbulence, equilibrium or non-

equilibrium, with focuses on wall friction and statistics of mean ﬂow and turbulence. See

reviews provided by Raupach et al. (1991), Jim´enez (2004), Chung et al. (2021), as well

as Finnigan (2000) in atmospheric applications. In addition, recent reviews on current

knowledge regarding rough-wall non-equilibrium turbulence were provided by Devenport

& Lowe (2022) and Volino et al. (2022).

One approach to describe and quantify the dynamical eﬀects of roughness on turbulence

is to analyze changes in the governing equations due to the presence of roughness. As

shown by Raupach & Shaw (1982), Mignot et al. (2009) and Yuan & Piomelli (2015),

additional terms in the transport equations of linear momentum and Reynolds stresses

appear with the presence of roughness when a double-averaging process is applied to the

equations. This averaging process performs a triple decomposition of an instantaneous

ﬂuid variable θ(x, y, z, t), such that θ=〈θ〉+!

θ+θ′, where () is time averaging for a

steady problem (or ensemble averaging for an unsteady one), ()′= () −() is turbulent

ﬂuctuation, and 〈〉 is intrinsic spatial averaging (i.e. value per unit ﬂuid area) along

homogeneous directions. !

θ=θ− 〈θ〉is the spatial heterogeneity of the local averaged θ,

called the dispersive or form-induced ﬂuctuations. Superﬁcial spatial averaging 〈·〉s(i.e.

ﬂuid variable per unit total area) is also used. 〈θ〉is the double-averaged (DA) value.

†Department of Mechanical Engineering, Michigan State University

‡Corresponding author. Email: junlin@egr.msu.edu

2Mangavelli & Yuan

The DA linear momentum equation is

∂〈ui〉s

∂t+∂

∂xj

〈ui〉s〈uj〉=−1

ρ

∂〈p〉s

∂xi

+ν∂2〈ui〉s

∂xj2−∂〈u′

iu′

j〉s

∂xj

−∂〈!ui!uj〉s

∂xj

+fi.(1.1)

Roughness gives rise to two additional terms in Equation (1.1): the fourth term on

the right-hand-side representing the momentum transfer due to form-induced stresses,

〈!ui!uj〉s(Nikora et al. (2001), also termed “dispersive stresses” by Wilson & Shaw (1977)),

and a body force (fi) representing the sum of local pressure drag and viscous stress

imposed on the surface of roughness. These two terms are signiﬁcant near the wall.

The near-wall layer in which the form-induced stress is signiﬁcant compared to the DA

velocity is called the roughness sublayer (RSL) (Pokrajac et al. 2007). Jelly & Busse

(2019) showed that both 〈!u!v〉and 〈u′v′〉, in opposite signs, contribute signiﬁcantly to the

mean momentum balance near the roughness crest for a Gaussian roughness. Mangavelli

et al. (2021) showed that, although in a fully developed channel ﬂow 〈!u!v〉is much weaker

than 〈u′v′〉, in a transient channel ﬂow the 〈!u!v〉magnitude may exceed temporarily that

of 〈u′v′〉below the roughness crest.

The DA equations of normal Reynolds stresses (considering a channel ﬂow) can be

written as (no summation over Greek index),

∂〈u′2

α〉s

∂t="

#

#

#

$−2〈u′

αv′〉s

∂〈uα〉

∂y

% &' (

Ps

+Pw+Pm

% &' (

Pf

s

)

*

*

*

+−,∂

∂xj

!

u′

αu′

α!uj-s

% &' (

Tw

−,∂

∂xj

u′

αu′

αu′

j-s

% &' (

Tt

−2,P′∂u′

α

∂xα-s

% &' (

Π

−2∂〈P′u′

α〉s

∂xα

% &' (

Tp

+ν.∂2u′2

α

∂xj∂xj/s

% &' (

Tν

−2ν,∂u′

α

∂xj

∂u′

α

∂xj-s

% &' (

"

,(1.2)

where the terms on the right-hand-side are, respectively, shear production due to double-

averaged strain rates (Ps), additional shear production due to form-induced strain rates

(Pf

s), additional transport due to form-induced velocities (Tw), turbulent transport (Tt),

viscous transport (Tν), pressure-strain-rate term (Π), pressure transport (Tp), and viscous

dissipation (%). Discussions on how the form-induced velocity modiﬁes Reynolds-stress

balance focused mainly on the Pf

sterm, which represents conversion from the kinetic

energy of the form-induced ﬂuctuations at the roughness length scale (or “wake kinetic

energy”, WKE (Raupach & Shaw 1982)) to the turbulent kinetic energy (TKE) at smaller

scales. The form-induced production was found to depend on the roughness geometry

(Yuan & Aghaei-Jouybari 2018) and to play a more important role in Reynolds stress

balance in non-equilibrium ﬂows, such as in spatially accelerating boundary layers (Yuan

& Piomelli 2015), oscillatory channel (Ghodke & Apte 2016) and transient accelerating

channels (Mangavelli et al. 2021), than in canonical ﬂows (Mignot et al. 2009; Yuan &

Piomelli 2014c). In these studies, Twwas found to be small in comparison to other terms.

Besides TKE production, another key process that controls the development of tur-

bulence is the distribution of TKE among velocity ﬂuctuations in diﬀerent directions,

represented by the pressure strain term in Equation (1.2) and partially controlled by

local p′intensity. It is not yet clear whether and how form-induced velocity modiﬁes

turbulent pressure and TKE redistribution.

Rough-wall turbulent ﬂows in engineering applications such as those over naval vessels

Eﬀects of form-induced velocity in turbulent channel ﬂows 3

or airfoils/hydrofoils usually display temporal and/or spatial variations due to external

factors including freestream pressure gradient, surface curvature, and ﬂow unsteadiness.

These ﬂows are termed non-equilibrium if a similarity solution of velocity cannot be

found. One type of non-equilibrium turbulence widely studied is the one under accelera-

tion (or favorable longitudinal pressure gradient, FPG). Studies on smooth walls showed

that a strong freestream acceleration causes the boundary layer to undergo reverse tran-

sition from a fully turbulent state to a quasi-laminar state (Narasimha & Sreenivasan

1973). This was attributed to the stretching of near-wall turbulent eddies, leading to elon-

gated velocity streaks, quasi-one-dimensional turbulence, reduced TKE redistribution,

and consequently stabilized turbulence, whose intensity decouples from the accelerated

mean ﬂow (Bourassa & Thomas 2009; Piomelli & Yuan 2013; Volino 2020). Roughness

acts to counter the stabilizing eﬀects of FPG (Cal et al. 2009; Yuan & Piomelli 2015), as

it augments near-wall TKE and leads to a more isotropic Reynolds stress tensor.

Transient accelerated ﬂows were shown to be fundamentally similar to those that un-

dergo spatial acceleration. Based on DNS, He & Seddighi (2013) and Seddighi et al.

(2015) characterized the response of turbulent channel ﬂows to an impulse-like increase

in ﬂow rate, over smooth and rough walls. On the smooth wall, they observed similar ﬂow

response as that in a FPG boundary layer undergoing reverse transition: elongated low-

speed streaks, laminar-like mean velocity proﬁles, higher Reynolds stress anisotropy, and

a frozen pressure strain term. In the presence of pyramid roughness, such reverse transi-

tion was prevented. Based on DNS of transient half-channel ﬂows with a similar conﬁgu-

ration, Mangavelli et al. (2021) characterized how two diﬀerent roughnesses with similar

average and root-mean-square heights but diﬀerent geometries aﬀect the evolutions of

mean and turbulent statistics. !uiin the RSL was observed to stay quasi-equilibrium

(scaling with the velocity at the edge of the sublayer) throughout the transient process,

in stark contrast to the non-equilibrium variation of u′

i. Strong form-induced production

of TKE in all u′

icomponents contributes to a much faster recovery of the steady state

on rough walls than on the smooth wall. The roughness geometry determines how fast

turbulence recovers to the steady state.

The description above summarizes the current knowledge on how roughness, as well

as the !uiﬁelds it induces, dynamically aﬀects equilibrium and non-equilibrium wall-

bounded turbulent ﬂows. However, a few questions are still unanswered. One is whether

form-induced velocity meaningfully aﬀects Reynolds stress balance in a way other than

additional TKE production, for example in modifying turbulent pressure and TKE redis-

tribution. In addition, most previous attention were given to eﬀects on turbulent statis-

tics. Does !uialso contribute to diﬀerent characteristics of turbulent structure observed

on diﬀerent rough walls?

2. Objectives

This work investigates the above questions in equilibrium and non-equilibrium channel

ﬂows, based on existing DNS data (Mangavelli et al. 2021) of transient half-channel ﬂows

subject to impulse accelerations on a smooth wall and two rough walls: a homogeneous,

densely distributed sandgrain roughness and an inhomogeneous, multiscale roughness

obtained from a hydraulic turbine scan. Mangavelli et al. (2021) focused on the devel-

opment of turbulent statistics, while the present work extends the analyses to turbulent

structure and various pathways through which !uiaﬀects turbulence.

The organization of the rest of the manuscript is as follows. In Sec. 3, simulation

4Mangavelli & Yuan

Case Reb0Reτ0k+

s∞,0(∆x+,∆z+)0Reb1Reτ1k+

s∞,1(∆x+,∆z+)1

SM 4,000 244 0 (4.0, 2.0) 12,000 626 0 (9.8, 4.8)

S 4,000 320 21 (5.0, 2.5) 12,000 1023 76 (15.3, 7.6)

T 4,000 294 10 (3.4, 3.4) 12,000 858 23 (10.1, 10.1)

Table 1: Simulation parameters at initial (‘0’) and new (‘1’) steady states. ‘+’ indicates

normalization in wall units (i.e. uτand viscous length scale δν=ν/uτ). ∆xiis the cell

size in xidirection. Reb=ubδ/νand Reτ=uτδ/ν.δis channel half height.

methodologies and parameters are described; statistical comparisons of the ﬂow are also

summarized. Then, two-point velocity correlations and spectra are compared in both

steady and transient states in Sec. 5.1. To provide an explanation to the structural

diﬀerence, the rate-of-strain tensor of !uiis characterized in Sec. 5.2. The velocity sources

of the turbulent pressure are compared in Sec. 5.3 to evaluate the eﬀects of 〈ui〉,!uiand

u′

ion pressure ﬂuctuations. Conclusions are provided in Sec. 6.

3. Problem fomulation

A summary of the methodologies and simulation setups as reported in Mangavelli et al.

(2021) is provided here. The readers are referred to that publication for further details.

The incompressible ﬂow of a Newtonian ﬂuid was simulated by solving the equations

of conservation of mass and momentum. x,yand zare, respectively, the streamwise,

wall-normal and spanwise directions of the half-channel ﬂow, and u,vand ware the

velocity components in those directions. pis the pressure, ρis the density and νis

the kinematic viscosity. Periodic boundary conditions are applied in xand zdomain

boundaries, and symmetric and no-slip conditions are applied to the top and bottom

boundaries, respectively. In the rough cases, an immersed boundary method was used

to impose the no-slip and no-penetration boundary condition on the rough walls. It is

based on the volume-of-ﬂuid approach; its detailed implementation in the in-house ﬂuid

solver was described by Yuan & Piomelli (2014c) and Yuan & Piomelli (2014b). The

governing equations were solved on a staggered grid using second-order central diﬀerences

for all terms, second-order accurate Adams-Bashforth semi-implicit time advancement,

and Message-Passing Interface parallelization.

To conduct the present transient channel simulations, ﬁrst a separate simulation was

carried out for each case at the initial steady state with a bulk Reynolds number Reb0,

to generate statistics at this state and be used as initial conditions for the transient

simulations. The transient simulations were then carried out, based on independent initial

conditions, for around 20 times for each case to collect data for ensemble averaging at each

time t. During a transient simulation, the variation of Reb(t) was imposed, equivalent

to a rapid three-fold linear increase of the bulk velocity (ub) that started at time t= 0

and lasted for a duration of t∗=tub0/δ= 0.08 (see Figure 2a). ubremained constant

thereafter. This simulation setup is similar to that of He & Seddighi (2013), though with

quantitative diﬀerences in the Reynolds number and the amount and rate of the velocity

ramp-up.

Simulation parameters for all cases are listed in Table 1. The initial and ﬁnal steady

states are denoted using subscripts “0” and “1”, respectively. “SM”, “S”, and “T” repre-

sent the smooth-wall, sandgrain, and a multiscale turbine-blade-roughness cases, respec-

Eﬀects of form-induced velocity in turbulent channel ﬂows 5

Figure 1: (a) Geometries of sandgrain (case S) and turbine-blade-roughness (case T)

surfaces colored by height, with zoomed-in view. (b) Power spectral density of height

ﬂuctuations, with wavenumber κin x(black) and z(gray). (c) Sketch of a rough surface

showing various length deﬁnitions.

tively. The same time-variation of Rebwas imposed in all cases. Both rough cases started

in transitionally rough regime at the initial steady state and became fully rough at the

new steady state, as shown by the k+

s∞values. Here, ks∞is the equivalent Nikuradse

sandgrain height in the fully rough regime of a rough surface; the critical values of k+

s∞

for the present roughnesses corresponding to the lower limit of the fully rough regime

were reported by Yuan & Piomelli (2014a).

On a rough wall, the wall shear stress τwresults from both the viscous and pressure

drag due to roughness. It was calculated as

τw(t) = ρ

LxLz0V

f1(x, y, z, t) dxdydz+µ∂〈u〉s

∂y1111y=0

,(3.1)

6Mangavelli & Yuan

Surface Ra/δkc/δkrms/RaskkuESxESzyR/kc

S 0.014 0.09 1.05 0.48 2.97 0.43 0.44 1.10 ±0.13

T 0.014 0.13 1.17 0.20 3.49 0.10 0.08 1.05 ±0.10

Table 2: Characteristics of the rough surfaces.

where f1(x, y, z, t) is the streamwise component of the immersed boundary method body

force, Vis the volume of the simulation domain, and Lxiis the domain length in xidirec-

tion. For the two rough surfaces considered herein, the second term on the right-hand-side

of Equation (3.1) is negligible. The friction velocity was then obtained as uτ(t) = 2τw/ρ.

The corresponding friction Reynolds numbers are tabulated in Table 1; the sandgrain

roughness (case S) produced higher wall friction than the multiscale roughness (case T).

Grid sizes of the uniform mesh in xand zare shown in wall units in Table 1. In y, the

grid is reﬁned near the wall. At the ﬁnal steady state (i.e. the more critical one for spatial

resolution between the two steady states), ∆y+

min <0.3 for SM and between 0.6 and 1.7

in the layer below roughness crest for the rough cases.

Two diﬀerent surfaces roughness, one synthetic sandgrain roughness ‘S’ and one repli-

cated from a surface scan on a hydraulic turbine blade ‘T’, are used in the simulations.

Figures 1(a) and (b) compare the roughness geometries and the power spectral density

of their height ﬂuctuations. Case S shows a prominent spectral peak at the separation

length between neighboring grains in xand z, denoted by λs, while the T roughness re-

sembles a fractal roughness with a spectral power decay of −2, without a dominant peak

at a particular length scale. For further discussion regarding the height power spectral

densities, see Mangavelli et al. (2021).

Parameters of the two rough surfaces are compared in Table 2. Both surfaces share the

same ﬁrst-order (Ra) and similar second-order moments (krms) of height statistics, but

quite diﬀerent values of maximum peak-to-trough height (kc, also called crest height),

skewness (sk) and eﬀective slope (ES) in xor z. Both surfaces are positively skewed (i.e.

peaky), with kurtosis values of around 3. Compared to S, the T roughness displays sparser

distribution of peaks, leading to a lower skewness value and milder eﬀective slopes. The

thickness of the RSL (i.e. yR, deﬁned as the layer in which 〈!u2〉1/2(y)/〈u〉(y)>0.1) is

around 1.1 times of kcin both cases. yRvaries mildly (within around 10%) throughout the

transient process. The domain lengths in xand zare 12δand 6δ, respectively, for cases

SM and S, and both 12δin case T to accommodate its large in-plane roughness length

scales in both xand z. For case S, the spatial resolution of the grain geometry is 8 and

16 points per grain in xand z, respectively. For case T, there is no dominant roughness

length scale. The Taylor micro-scale (λT) (Yuan & Piomelli 2014a; Aghaei Jouybari

et al. 2021) is thus used to quantify the length of an equivalent roughness element. λTis

resolved by around 10 points in both xand z.

4. Summary of turbulence statistics

Some observations on ﬂow statistics are summarized here to provide a context for the

structural and other analyses reported in Section 5. For comprehensive discussions of the

ﬂows, see Mangavelli et al. (2021).

Figure 2(a) shows the acceleration of ubthat was imposed in all cases. The friction co-

Eﬀects of form-induced velocity in turbulent channel ﬂows 7

Figure 2: Flow statistics of the transient channels: (a) bulk velocity, (b) friction coeﬃcient,

and wall-normal maximum values of streamwise and vertical normal Reynolds stresses

(c,d) and form-induced stresses (e,f). (b-f) are reproduced from Mangavelli et al. (2021)

with permission.

eﬃcient Cf= 2τw(t)/(ρu2

b0) is compared in Figure 2(b). Cfundergoes a sudden increase

for all cases immediately following the ubramp-up, and decreases rapidly as the ﬂow

starts to recover. Figures 2(c-f) highlight the diﬀerences in Reynolds and form-induced

stress developments in the three cases.

On the smooth wall, turbulence undergoes reverse transition toward a quasi-laminar

state, characterized by a dip of Cfbelow its long-time value and high Reynolds stress

anisotropy (with TKE mostly resting in u′), caused by a delayed response of the TKE

redistribution to the acceleration. On the rough walls, however, a clear dip of Cfis absent,

8Mangavelli & Yuan

Figure 3: Contours of R11 at levels 0.3(0.2)0.9 in Cases SM (a), S (b) and T(c), at four

time instances: t∗= 0.0, 0.5, 5, and 22. Lines ﬁtted based on outermost points (an

example is shown with ◦in (a)) at contour levels 0.5(0.1)0.9, to show inclination angles

(marked in red). For clarity, a shift of 4.5 and 1.5 units are used for smooth and rough

cases, respectively.

and Reynolds stress anisotropy stays almost unchanged. As opposed to the monotonic

increase of u′

iﬂuctuations to a higher-Reynolds-number state, the !uiﬂuctuations display

rapid augmentation following the ubramp-up and then decrease toward the long-time

values. This variation of !uiintensity was shown to be a result of the variation of 〈u〉at

the edge of the RSL; the ratio between the root-mean-square (RMS) of !uiand 〈u〉(yR)

was shown to be almost constant for each rough case.

5. Results

5.1. Structural characteristics of turbulent velocity

First, the two-point auto-correlation of the turbulent velocity u′

icentered at an elevation

yref is calculated as

R11(rx, ry, t) = 〈u′(x, yref , z, t)u′(x+rx, yref +ry, z, t)〉

〈u′2〉(yref , t),(5.1)

where rxand ryare separations in streamwise and wall-normal directions, respectively.

The intrinsic averaging operator in the nominator on the right-hand-side of Equation (5.1)

indicates that only contributions from products with both points (*xand *x+*r) in the

ﬂuid domain are accounted for in calculating the correlation. Contours of R11 in the

(x, y) plane are shown in Figures 3 for all cases at t∗= 0.0, 0.5, 5, and 22. In the smooth

case, the correlation is centered at y+

ref,0=uτ0yref/ν= 10, which corresponds to the

peak elevation of the TKE production at t∗= 0, while in the rough cases yref/kc= 1,

which is near the peak elevations of normal Reynolds stresses (Mangavelli et al. 2021).

Following Volino et al. (2007), the inclination angle θis calculated using the best ﬁtted

line by the linear least square method, based on the points farthest away from the self-

correlation point at (rx, ry) = (0,0), both upstream and downstream on the contour lines

Eﬀects of form-induced velocity in turbulent channel ﬂows 9

Figure 4: Temporal variation of streamwise integral length (a,c) and inclination angle

(b,d) based on R11 in SM (◦) , S (△), T ( ) cases. The correlations are centered at

y+

ref,0= 10 for SM and yref/kc= 1 for S and T. (a,c) show linear scalings of t∗and (c,d)

show logarithmic scalings.

with levels from 0.5 to 0.9. These outermost points used for line ﬁtting are shown in the

R11 contours at t∗= 0.0 in Figure 3(a), as an example. The values of θare listed in the

ﬁgure. To compare quantitatively the size and inclin